What's new inPhysics as a Generalized Number TheoryNote: Newest contributions are at the top! 
Year 2013 
Riemann Hypothesis and quasicrystalsFreeman Dyson has represented a highly interesting speculation related to Riemann hypothesis and 1dimensional quasicrystals (QCs). He discusses QCs and Riemann hypothesis briefly in his Einstein lecture. Dyson begins from the defining property of QC as discrete set of points of Euclidian space for which the spectrum of wave vectors associated with the Fourier transform is also discrete. What this says is that quasicrystal as also ordinary crystal creates discrete diffraction spectrum. This presumably holds true also in higher dimensions than D=1 although Dyson considers mostly D=1 case. Thus QC and its dual would correspond to discrete points sets. I will consider the consequences in TGD framework below. Dyson considers first QCs at general level. Dyson claims that QCs are possible only in dimensions D=1,2,3. I do not know whether this is really the case. In dimension D=3 the known QCs have icosahedral symmetry and there are only very few of them. In 2D case (Penrose tilings) there is nfold symmetry, roughly one kind of QC associated with any regular polygon. Penrose tilings correspond to n=5. In 1D case there is no point group (subgroup of rotation group) and this explains why the number of QCs is infinite. For instance, so called PV numbers identified as algebraic integers, which are roots of any polynomial with integer coefficients such that all other roots have modulus smaller than unity. 1D QCs is at least as rich a structure as PV numbers and probably much richer. Dyson suggests that Riemann hypothesis and its generalisations might be proved by studying 1D quasicrystals.

pAdic symmetriesThe recent progress in the formulation of the notion of padic manifold is so important for the program of defining quantum TGD in mathematically rigorous manner that it deserves a series of more detailed postings devoted to the notion of padic manifold, padic integration, and padic symmetries. This posting is the third one and devoted to padic symmetries. A further objection relates to symmetries. It has become already clear that discrete subgroups of Liegroups of symmetries cannot be realized padically without introducing algebraic extensions of padics making it possible to represent the padic counterparts of real group elements. Therefore symmetry breaking is unavoidable in padic context: one can speak only about realization of discrete subgroups for the direct generalizations of real symmetry groups. The interpretation for the symmetry breaking is in terms of discretization serving as a correlate for finite measurement resolution reflecting itself also at the level of symmetries. This observation has led to TGD inspired proposal for the realization of the padic counterparts symmetric spaces resembling the construction of P^{1}(K) in many respects but also differing from it.
The Lie algebra of the rotation group spanned by the generators L_{x},L_{y},L_{z} provides a good example of the situation and leads to the question whether the hierarchy of Planck constants kenociteallb/Planck could be understood padically.
Canonical identification and the definition of padic counterparts of Lie groups For Lie groups for which matrix elements satisfy algebraic equations, algebraic subgroups with rational matrix elements could regarded as belonging to the intersection of real and padic worlds, and algebraic continuation by replacing rationals by reals or padics defines the real and padic counterparts of these algebraic groups. The challenge is to construct the canonical identification map between these groups: this map would identify the common rationals and possible common algebraic points on both sides and could be seen also a projection induced by finite measurement resolution. A proposal for a construction of the padic variants of Lie groups was discussed in previous section. It was found that the padic variant of Lie group decomposes to a union of disjoint sets defined by a discrete subgroup G_{0} multiplied by the padic counterpart G_{p,n} of the continuous Lie group G. The representability of the discrete group requires an algebraic extension of padic numbers. The disturbing feature of the construction is that the padic cosets are disjoint. Canonical identification I_{k,l} suggests a natural solution to the problem. The following is a rough sketch leaving a lot of details open.
For details see the new chapter What padic icosahedron could mean? And what about padic manifold? or the article with the same title. 
Could canonical identification make possible definition of integrals in padic context?The recent progress in the formulation of the notion of padic manifold is so important for the program of defining quantum TGD in mathematically rigorous manner that it deserves a series of more detailed postings devoted to the notion of padic manifold, padic integration, and padic symmetries. This posting is the second one and devoted to padic integration. The notion of padic manifold using using real chart maps instead of padic ones allows an attractive approach also to padic integration and to the problem of defining padic version of differential forms and their integrals.
For details see the new chapter What padic icosahedron could mean? And what about padic manifold? or the article with the same title. 
Could canonical identification allow construction of path connected topologies for padic manifolds?The recent progress in the formulation of the notion of padic manifold is so important for the program of defining quantum TGD in mathematically rigorous manner that it deserves a series of more detailed postings devoted to the notion of padic manifold, padic integration, and padic symmetries. This posting is the first one and devoted to the notion of padic manifold. Total disconnectedness of padic numbers as the basic problem The total disconnectedness of padic topology and lacking correspondence with real manifolds could be seen as genuine problem in the purely formal construction of padic manifolds. Physical intuition suggests that path connected should be realized in some natural manner and that one should have a close connection with real topology which after all is the "lab topology". In TGD framework one of the basic physical problems has been the connection between padic numbers and reals. Algebraic and topological approaches have been competing also here.
What about padic coordinate charts for a real preferred extremal? What is remarkable that one can also build padic coordinate charts about real preferred extremal using the inverse of the canonical identification assuming that finite rationals are mapped to finite rationals. There are actually good reasons to expect that coordinate charts make sense in both directions. Algebraic continuation from real to padic context is one such reason. At the real side one can calculate the values of various integrals like K ähler action. This would favor padic regions as map leafs. One can require that K ähler action for Minkowskian and Euclidian regions (or their appropriate exponents) make sense padically and define the values of these functions for the padic preferred extremals by algebraic continuation. This could be very powerful criterion allowing to assign only very few padic primes to a given real spacetime surface. This would also allow to define padic boundaries as images of real boundaries in finite measurement resolution. pAdic path connectedness would be induced from real pathconnectedness. pAdic rationals include also the ratios of integers, which are infinite as real integers so that the pinary expansion of the rational is not periodic asymptotically. In principle one could imagine of mapping also these to real numbers but the resulting skeleton might be too dense and might not allow to satisfy the preferred extremal property. Furthermore, the representation of a padic number as a ratio of this kind of integers is not unique and can be always tranformed to an infinite padic integer multiplied by a power of p . In the same manner real points which can be regarded as images of ratios of padic integers infinite as real integers could be mapped to padic ones but same problem is encountered also now. In the intersection of real and padic worlds the correspondence is certainly unique and means that one interprets the equations defining the padic spacetime surface as real equations. The number of rational points (with cutoff) for the padic preferred extremal becomes a measure for how unique the chart map in the general case can be. For instance, for 2D surfaces the surfaces x^{n}+y^{n}=z^{n} allow no nontrivial rational solutions for n>2 for finite real integers. This criterion does not distinguish between different padic primes and algebraic continuation is needed to make this distinction. Chart maps for padic manifolds The real map leafs must be mutually consistent so that there must be maps relating coordinates used in the overlapping regions of coordinate charts on both real and padic side. On padic side chart maps between real map leafs are naturally induced by identifying the canonical image points of identified padic points on the real side. For discrete chart maps I_{k,l} with finite pinary cutoffs one one must complete the real chart map to  say diffeomorphism. That this completion is not unique reflects the finite measurement resolution. In TGD framework the situation is dramatically simpler. For submanifolds the manifold structure is induced from that of imbedding space and it is enough to construct the manifold structure M^{4} × CP_{2} in a given measurement resolution (k,l). Due to the isometries of the factors of the imbedding space, the chart maps in both real and padic case are known in preferred imbedding space coordinates. As already discussed, this allows to achieve an almost complete general coordinate invariance by using subset of imbedding space coordinates for the spacetime surface. The breaking of GCI has interpretation in terms of presence of cognition and selection of quantization axes. For instance, in the case of Riemann sphere S^{2} the holomorphism relating the complex coordinates in which rotations act as M öbius tranformations and rotations around call it zaxis act as phase multiplications  the coordinates z and w at Norther and Southern hemispheres are identified as w=1/z restricted to rational points at both side. For CP_{2} one has three poles instead of two but the situation is otherwise essentially the same. For details see the new chapter What padic icosahedron could mean? And what about padic manifold? or the article with the same title. 
What padic icosahedron could mean? And what about padic manifold?I have been working for a couple of weeks with the problem of defining the notion of padic manifold: this is one of the key challenges of TGD. The existing proposals by mathematicians are rather complicated and it seems that something is lacking. To my opinion, to identify this something it is essential to make the question "What padic numbers are supposed to describe?". This question has not bothered either matheticians or theoretical physicists proposing purely formal padic counterparts for the scattering amplitudes. Without any answer to this question there are simply quite too many alternatives to consider and one ends up to the garden of branching paths. The text below is this introduction to the article and chapter about the topics.  This article was originally meant to be a summary of what I understand about the article "The pAdic Icosahedron" in Notices of AMS. The original purpose was to summarize the basic ideas and discuss my own view about more technical aspects  in particular the generalization of Riemann sphere to padic context which is rather technical and leads to the notion of Bruhat Tits tree and Berkovich space. About BruhatTits tree there is a nice web article titled pAdic numbers and BruhatTits tree describing also basics of padic numbers in a very concise form. The notion of padic icosahedron leads to the challenge of constructing padic sphere, and more generally padic manifolds and this extended the intended scope of the article and led to consider the fundamental questions related to the construction of TGD. Quite generally, there are two approaches to the construction of manifolds based on algebra resp. topology.
The attempt to construct padic manifolds by mimicking topological construction of real manifolds meets difficulties The basic problem in the application of topological method to manifold construction is that padic disks are either disjoint or nested so that the standard construction of real manifolds using partially overlapping nballs does not generalize to the padic context. The notions of BruhatTits tree, building, and Berkovich disks and Berkovich space are represent attempts to overcome this problem. Berkovich disk is a generalization of the padic disk obtained by adding additional points so that the padic disk is a dense subset of it. Berkovich disk allows path connected topology which is not ultrametric. The generalization of this construction is used to construct padic manifolds using the modification of the topological construction in the real case. This construction provides also insights about padic integration. The construction is highly technical and complex and pragmatic physicist could argue that it contains several unnatural features due to the forcing of the real picture to padic context. In particular, one must give up the padic topology whose ultrametricity has a nice interpretation in the applications to both padic mass calculations and to consciousness theory. I do not know whether the construction of BruhatTits tree, which works for projective spaces but not for Q_{p}^{n} (!) is a special feature of projective spaces, whether BruhatTits tree is enough so that no completion would be needed, and whether BruhatTits tree can be deduced from Berkovich approach. What is remarkable that for M^{4}× CP_{2} padic S^{2} and CP_{2} are projective spaces and allow BruhatTits tree. This not true for the spheres associated with the lightcone boundary of D≠ 4dimensional Minkowski spaces. Two basic philosophies concerning the construction of padic manifolds There exists two basic philosophies concerning the construction of padic manifolds: algebraic and topological approach. Also in TGD these approaches have been competing: algebraic approach relates real and padic spacetime points by identifying common rationals. Finite pinary cutoff is however required to achieve continuity and has interpretation in terms of finite measurement resolution. Canonical identification maps padics to reals and vice versa in a continuous manner but is not consistent with field equations without pinary cutoff.
Number theoretical universality and the construction of padic manifolds Construction of padic counterparts of manifolds is also one of the basic challenges of TGD. Here the basic vision is that one must take a wider perspective. One must unify real and various padic physics to single coherent whole and to relate them. At the level of mathematics this requires fusion of real and padic number fields along common rationals and the notion of algebraic continuation between number fields becomes a basic tool. The number theoretic approach is essentially algebraic and based on the gluing of reals and various padic number fields to a larger structure along rationals and also along common algebraic numbers. A strong motivation for the algebraic approach comes from the fact that preferred extremals are characterized by a generalization of the complex structure to 4D case both in Euclidian and Minkowskian signature. This generalization is independent of the action principle. This allows a straightforward identification of the padic counterparts of preferred extremals. The algebraic extensions of padic numbers play a key role and make it possible to realize the symmetries in the same manner as they are realized in the construction of padic icosahedron. The lack of wellordering of padic numbers implies strong constraints on the formulation of number theoretical universality.
How to achieve path connectedness? The basic problem in the construction of padic manifolds is the total disconnectedness of the padic topology implied by ultrametricity. This leads also to problems with the notion of padic integration. Physically it seems clear that the notion of path connectedness should have some physical counterpart. The notion of open set makes possib le path connectedness possible in the real context. In padic context BruhatTits tree and Berkovich disk are introduced to achieve the same goal. One can of course ask whether Berkovich space could allow to achieve a more rigorous formulation for the padic counterparts of CP_{2}, of partonic 2surfaces, their lightlike orbits, preferred extremals of Kähler action, and even the "world of classical worlds" (WCW). To me this construction does not look promising in TGD framework but I could be wrong. TGD suggests two alternative approaches to the problem of path connectedness. They should be equivalent. pAdic manifold concept based on canonical identification The TGD inspired solution to the construction of path connectd padic topology is based on the notion of canonical identification mapping reals to padics and vice versa in a continuous manner.
Could path connectedness have quantal description? The physical content of path connectedness might also allow a formulation as a quantum physical rather than primarily topological notion, and could boil down to the nontriviality of correlation functions for second quantized induced spinor fields essential for the formulation of WCW spinor structure. Fermion fields and their npoint functions could become part of a number theoretically universal definition of manifold in accordance with the TGD inspired vision that WCW geometry  and perhaps even spacetime geometry  allow a formulation in terms of fermions. The natural question of physicist is whether quantum theory could provide a fresh number theoretically universal approach to the problem. The basic underlying vision in TGD framework is that second quantized fermion fields might allow to formulate the geometry of "world of classical worlds" (WCW) (for instance, Kähler action for preferred extremals and thus Kähler geometry of WCW would reduce to Dirac determinant. Maybe even the geometry of spacetime surfaces could be expressed in terms of fermionic correlation functions. This inspires the idea that second quantized fermionic fields replace the Kvalued (K is algebraic extension of padic numbers) functions defined on padic disk in the construction of Berkovich. The ultrametric norm for the functions defined in padic disk would be replaced by the fermionic correlation functions and different Berkovich norms correspond to different measurement resolutions so that one obtains also a connection with hyperfinite factors of type II_{1}. The existence of nontrivial fermionic correlation functions would be the counterpart for the path connectedness at spacetime level. The 3surfaces defining boundaries of a connected preferred extremal are also in a natural manner "path connected": the "path" is defined by the 4surface. At the level of WCW and in zero energy ontology (ZEO) WCW spinor fields are analogous to correlation functions having collections of these disjoint 3surfaces as arguments. There would be no need to complete padic topology to a path connected topology in this approach. It must be emphasized that this apporach should be consistent with the first option and that it is much more speculative that the first option. About literature It is not easy to find readable literature from these topics. The Wikipedia article about Berkovich space is written with a jargon giving no idea about what is involved. There are video lectures about Berkovich spaces. The web article about Berkovich spaces by Temkin seems too technical for a nonspecialist. The slides however give a concise bird's eye of view about the basic idea behind Berkovich spaces. Topics of the article The article was originally meant to discuss padic icosahedron. Although the focus was redirected to the notion of padic manifold  especially in TGD framework  I decided to keep the original starting point since it provides a concrete manner to end up with the deep problems of padic manifold theory and illustrates the group theoretical ideas.
For details see the new chapter What padic icosahedron could mean? And what about padic manifold? or the article with the same title. 